Fast Marching-Cubes-Style Volume Evaluation for Level Set Surfaces

Abstract

We present an efficient and accurate volume evaluation method for grid-based level sets that computes the volume of the implicitly represented shape(s) in a manner consistent with Marching Cubes surface reconstruction. We utilize rotational symmetry to combine redundant Marching Cubes cases and avoid explicitly forming local triangulations using efficient volume computation formulae for pyramids and truncated prisms wherever possible, thereby achieving a fast and compact implementation. We demonstrate that our method is more efficient than previous approaches while generating results that converge with second-order accuracy and are perfectly consistent with volumes calculated directly from explicit Marching Cubes meshes. We provide a full C++17 reference implementation to foster adoption of the proposed method (https://github.com/tetsuya-takahashi/MC-style-vol-eval).